where哪里 is a random variable with probability distribution是随机变量与概率分布 representing demand, each unit is sold for price代表需求，各单位销售价格 and purchased for price购买价格 , ， is the number of units stocked, and是储藏丰富的单位数， is the expectation operator .为期望算子 。 The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:最优库存量的预期利润最大化的报童的解决方案是：

Intuitively, this ratio, referred to as the critical fractile , balances the cost of being understocked (a lost sale worth直观地说，这样的比例，称为为临界分位数 ，结余的成本被understocked（丢失的出售价值 ) and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or ）的总成本，无论是的积压或understocked（积压成本，库存成本，或 so total cost is simply因此总的成本是根本 ). ）。

If如果 (ie the retail price is less than the purchase price), the numerator becomes negative. （即零售价低于购买价格），分子变为负数。 In this situation, it isn't worth keeping any item in the inventory.在这种情况下，这是不值得的库存中的任何项目。

Assuming that the 'newsvendor' is in fact a small company who wants to produce goods to an uncertain market.假设，“报童”，其实是一家小公司，谁愿意生产产品不明朗的市场。 In this more general situation the cost function of the newsvendor (company) can be formulated in the following manner:在这种更一般的情况下的的报童（公司）的成本函数可以以下面的方式配制：

where the individual parameters are the following:各个参数如下：

– fixed cost. - 固定的成本。 This cost always exists when the production of a series is started.开始时，总是存在着一系列的生产成本。 [$/production] [$ /生产] – variable cost. - 可变成本。 This cost type expresses the production cost of one product.此成本类型表示一个产品的生产成本。 [$/product] [$ /产品] – The product quantity in the inventory. - 产品的库存数量。 The decision of the inventory control policy concerns the product quantity in the inventory after the product decision.库存控制策略的决定，涉及的产品数量清单中的产品后决定。 This parameter includes the initial inventory as well.该参数包括初始库存为好。 If nothing is produced, then this quantity is equal to the initial quantity, ie concerning the existing inventory.如果没有产生，则此一量的初始量是相等的，即与现有的库存。 – Initial inventory level. - 初始库存水平。 We assume that the supplier possesses我们假设供应商拥有 products in the inventory at the beginning of the demand of the delivery period.产品清单中交付期开始时的需求。 – penalty cost (or back order cost). - 惩罚成本（或回订单成本）。 If there is less raw material in the inventory than needed to satisfy the demands, this is the penalty cost of the unsatisfied orders.如果有较少的原材料，在清单中的比，以满足需要的要求，这是未满足订单的惩罚成本。 [$/product] [$ /产品] – Expected value of the - 预期值 stochastic variable.随机变量。 – This means the demand from the receiver for the product, which is an optional probability variable. - 这意味着从接收器的需求的产品，这是一个可选的概率变量。 [unit] [单位] – inventory and stock holding cost. - 库存和库存持有成本。 [$ / product] [$ /产品]

On the basis of the cost function the determination of the optimal inventory level is a minimization problem.成本函数的基础上的最优存货水平的测定是一个最小化问题。 So in the long run the amount of cost-optimal end-product can be calculated on the basis of the following relation: [ 1 ]因此，在长期的运行成本最优的最终产品的量，可以计算出的基础上，下面的关系式： [1]